# Geometri och Topologi seminariet

Onsdagar kl 14.15 i sal Å64119.

Seminarieledare: Georgios Dimitroglou Rizell och Maksim Maydanskiy.

**Jeff Hicks**(Berkeley)

Room: Å64119

Title: Tropical Lagrangians and Mutations

Abstract: Mirror symmetry is a conjectured relation between the symplectic geometry of a space X and the complex geometry of a mirror space Y. A mechanism for this duality comes from the Strominger-Yau-Zaslow conjecture, which states that mirror spaces are dual Lagrangian torus fibrations over a common base Q. The connection between the symplectic geometry on X and complex geometry on Y is seen via a degeneration to tropical geometry on Q.

The recent work of Matessi and Mikhalkin show how to lift tropical curves in Q to Lagrangian submanifolds of X. I will provide a different construction of these tropical Lagrangians inspired by homological mirror symmetry, and explore how different Lagrangian lifts of these tropical curves may be related to each other by Lagrangian mutation.

Upcoming:

November 7 -- Tobias Ekholm.

Past:

Thomas Kragh (Uppsala)

September 26, 14:15

Room: Å64119

Title: Twisted generating families.

Abstract: This will be a report on work in progress with Abouzaid, Courte, and Guillermou. In this lecture I will discus a short proof of the existence of generating families for A Lagrangian embedding in a cotangent bundle with stable trivial Gauss map. I will then define the notion of a twisted generating family, and sketch why we think these exists/how we want to construct them. I will also discus what consequences this existence has.

Mohammed Abouzaid (Columbia)

May 3rd (THURSDAY), 15:15

Room: Å64119

Title: From flow categories to homotopy types

Abstract: A formal step in the construction of a Floer homotopy type is the passage from flow categories to stable homotopy types. Cohen-Jones-Segal gave a description of this via iterated Pontryagin-Thom constructions. I will explain a reformulation of this construction which is more amenable to considering general cohomology theories. This is joint work with A. Blumberg.

Wanmin Liu (IBS Center for Geometry and Physics)

April 25, 14:15

Room: Å64119

Title: Classification of full exceptional collections of line bundles on three blow-ups of P^3 and on some projective bundles.

Abstract: A fullness conjecture of Kuznetsov says that if a smooth projective variety X admits a full exceptional collection of line bundles of length l, then any exceptional collection of line bundles of length l is full. In this talk, we show that this conjecture holds for X as the blow-up of P^3 at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on X is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such X. The preprint is available at IBS-CGP preprint [CGP17025].

I will also introduce some recent progress on some projective bundles, including blow-up of P^n at a point. This is a joint work with Song Yang and Xun Yu at Tianjin University.

Johannes Nordstrom (University of Bath)

March 28, **15:15**

Room: **Å73121 Oseenska** (physics dept. sminar room).

Title: Disconnecting the G_2-moduli space

Abstract: A natural question in the study of Riemannian 7-manifolds with holonomy G_2 is whether the moduli space on a closed 7-manifold can be disconnected, i.e. whether there can exist different metrics of holonomy G_2 on the same closed manifold such that one cannot be deformed to the other through a path of G_2 metrics. Any holonomy G_2 metric has an associated G_2 structure, which satisfies a certain partial differential equation. I will present topological and analytic invariants of G_2-structures (largely ignoring the PDE), and examples where these invariants are able to distinguish components of the G_2 moduli space.

Georgios Dimitroglou Rizell (Uppsala)

February 28, 14:15

Room: Å64119

Title: Curve counts on Lagrangian immersions and multiplicative preprojective algebras

Abstract: We define open Gromov-Witten invariants with an immersed Lagrangian surface as boundary condition. It turns out that the count should be considered as an element inside a multiplicatively preprojective algebra. (In the embedded case this simply means the group ring of the fundamental group.) We explain these notions, and show how the result can be used to distinguish immersed Lagrangian spheres with one double point inside the projective planes up to Hamiltonian isotopy. This is joint work with T. Ekholm and D. Tonkonog.

Thomas Kragh (Uppsala)

January 17, **11:00**

Room: Å64119

Title: Generating families for exact Lagrangians with trivial stable Gauss map.

Abstract: Giroux proved that any Lagrangian embedding has a generating family if and only if its stable Gauss map is trivial. However, the type of generating family he considered may not have well-defined Morse theory (in the sense that there is no control over the compactness of gradient trajectories). In this talk I will sketch a proof that all Lagrangians with stably trivial Gauss map actually has a generating family with fiberwise compact set of gradient trajectories (this is very similar to having a g.f. quadratic at infinity).

January 10, **13:00 - 15:30**:

Jean-François Barraud (Toulouse)

Title: A Novikov fundamental group.

Abstract: Just like there might be homotopical constraints on the critical points of Morse functions that are not detected by the homology of the ambient manifold, closed 1 forms may have necessary critical points that are not detected by the Novikov homology. I will present an agebraic invariant that captures some of this information, and is an analogue in Novikov theory of the fundamental group in Morse theory. In particular, this "Novikov fundamental group" leads to new lower bounds for the number of index 1 and 2 critical points of closed 1-forms, that are essentially different from the classical Morse-Novikov inequalities. (jw with A. Gadbled and H.V.Le).

Thomas Kragh (Uppsala)

Title: On the space of Legendrians isotopic to the zero section in Jet-1 bundles.

## Tidigare års seminarier

## 2017

### Fall:

Seminarieledare: Georgios Dimitroglou Rizell och Maksim Maydanskiy.

Luigi Tizzano (Uppsala)

November 29, **10:15**

Room: Ångström **73121**

Title: Physics conjectures about topological Fukaya category.

Luis Diogo (Uppsala)

November 22, 14:15

Room: Ångström 3419

Title: Lifting Lagrangians from Donaldson-type divisors.

Abstract: Given a closed symplectic manifold with a symplectic submanifold of codimension 2, we can sometimes lift monotone Lagrangians from the submanifold to the ambient manifold. We show that under some assumptions, it is possible to write the superpotentials of the lifted Lagrangians in terms of the superpotentials of the original Lagrangians (we may also need additional information coming from relative Gromov-Witten invariants). The superpotential of a Lagrangian is a count of pseudoholomorphic disks (of Maslov index 2) with boundary on the Lagrangian, and it plays an important role in Floer theory and mirror symmetry.

We will discuss applications of this result, including how it can be used to distinguish infinitely many monotone Lagrangian tori in complex projective planes, quadrics and cubics of complex dimension at least 3. This is joint work with D. Tonkonog, R. Vianna and W. Wu.

Honghao Gao (Grenoble)

November 15, 14:15

Room: Ångström **3419**

Title: Augmentations and sheaves for knot conormals.

Abstract: Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. Nadler-Zaslow correspondence suggests a connection between the two types of invariants. Moreover, augmentations specialized to “Q=1” have been understood through KCH representations.

I will present a classification result of simple sheaves, and relate it to KCH representations and two-variable augmentation polynomials. I will also present a Radon transform for sheaf categories, and explain how it corresponds to the specialization of Q on the sheaf side.

Evgeny Volkov (Uppsala)

November 8, 14:15

Room: Ångström **3419**

Title: SFT without holomorphic curves.

Abstract: We construct a version of SFT for the cotangent bundle of a simply connected odd dimensional manifold without using holomorphic curves. There is a map from this SFT into string topology of the manifold. We show that this map induces an isomorphism on homology, and intertwines the SFT product with (a version of) the Chas-Sullivan product.

Paolo Ghiggini (Nantes)

October 25, 14:15

Room: Ångström 4004

Title: A dynamical regard on knot Floer homology.

Abstract: Knot Floer homology is a family of abelian groups \widehat{HFK}(Y, K, i) for a null-homologous knot K in a closed, oriented 3-manifold Y which is indexed by an integer i \in [-g, g], where g denotes the minimal genus of an embedded surface bounding K. This invariant was introduced by Ozsváth, Szabó and Rasmussen using a Lagrangian Floer homology construction. I will show that, when K is a fibred knot (i.e. Y-K fibres over S^{1} and K is the boundary of the closure of every fibre), the group \widehat{HFK}(Y, K, -g+1) is isomorphic to a version of the fix point Floer homology of any area-preserving representative of the monodromy of the fibration on Y-K. I will also discuss some potential applications of this isomorphism. This is a work in progress in collaboration with Gilberto Spano.

Roman Golovko (ULB)

October 25, 15:45

Room: Ångström 4004

Title: The wrapped Fukaya category of a Weinstein manifold is generated by the cocores of the critical Weinstein handles.

Abstract: We decompose any object in the wrapped Fukaya category of a 2n-dimensional Weinstein manifold as a twisted complex built from the cocores of the n-dimensional handles in a Weinstein handle decomposition. This is joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Paolo Ghiggini.

Sara Angela Filippini (Cambridge University)

October 18, 14:15

Room: Ångström **3419**

Title: Refined curve counting and the tropical vertex group

Abstract: The tropical vertex group of Kontsevich and Soibelman is generated by formal symplectomorphisms of the 2-dimensional algebraic torus. It plays a role in many problems in algebraic geometry and mathematical physics. Based on the tropical vertex group, Gross, Pandharipande and Siebert introduced an interesting Gromov-Witten theory on weighted projective planes which admits a very special expansion in terms of tropical counts. I will describe a refinement or "q-deformation" of this expansion, motivated by wall-crossing ideas, using Block-Göttsche invariants. This leads naturally to the definition of a class of putative q-deformed curve counts. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined. This is joint work with Jacopo Stoppa.

Yang Huang (Uppsala)

October 11, 14:15

Room: Ångström **3419**

Title: On convex hypersurface theory.

Abstract: In 3-dimensional contact topology, convex surface theory, developed by Giroux, plays a prominent role. In this talk, I will describe a ‘parallel’ story in higher dimensions. In particular I will introduce the notion of overtwisted orange and bypass as technical tools. This is based on joint work with K. Honda.

Tobias Ekholm (Uppsala)

October 4, 14:15

Room: Ångström 64119

Title: Skein relation from holomorphic curves.

Johan Asplund (Uppsala)

September 27, 14:15

Room: Ångström **4006**

Title: Flexibility of Legendrian immersions.

Abstract: Following Eliashberg and Mishachevs book, I will give a brief

overview of Gromovs famous h-principle and will discuss the fact that

Legendrian immersions satsifies the h-principle. Using the h-principle

I will then show that a closed n-dimensional submanifold has a

Legendrian immersion into standard contact R^{2n+1} if and only if the

complexified tangent bundle of the submanifold is trivial. Using this

fact, we will then exhibit surfaces that do not admit a Legendrian

immersion into R^{5} by computing some Chern classes.

### Spring:

Seminarieledare: Thomas Kragh och Maksim Maydanskiy.

Davide Alboresi (Utrecht)

May 31, **15:00**

Room: Ångström 64119

Title: Towards Fukaya categories of stable generalized complex manifolds.

Abstract: Stable generalized complex manifolds are a class of generalized complex manifolds which are symplectic almost everywhere, with a symplectic form that has a logarithmic singularity along a hypersurface. In this talk I will discuss an attempt to define a category of branes for such manifolds, mimicking the construction of the Wrapped Fukaya category.

Peter Feller (MPI Bonn)

May 24, 14:15

Room: Ångström 64119

Title: Knots with a view toward embedding problems in complex geometry.

Abstract: We first discuss classical questions about polynomial embeddings of the complex line C into complex spaces such as Cm and affine algebraic groups. Next, we consider torus knots and discuss questions related to their non-sliceness motivated by singularity theory. Finally, we use a knot theory perspective to indicate proofs for the embedding questions discussed first.

May 3, 14:15

Room: Ångström 64119

Title: Higher rank local systems for monotone Lagrangians.

Abstract: Lagrangian Floer homology is a tool for studying intersections of Lagrangian submanifolds of symplectic manifolds. It is defined by ``counting'' pseudoholomorphic discs with boundary on these submanifolds. One way to enrich this theory is to record some homotopy data about the paths that the boundaries of these discs trace on the Lagrangians. A simple algebraic way of accomplishing this is to use local coefficients. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not, one can always restrict to some natural unobstructed subcomplexes. I will showcase all of these constructions with some explicit calculations for the Chiang Lagrangian in CP3. Its Floer theory was computed by Evans and Likili, who also pointed out that standard Floer homology cannot tell us whether the Chiang Lagrangian and RP3 can be disjoined by a Hamiltonian isotopy. We will see how using a rank 2 local system in this example allows us to show that these two Lagrangians are in fact non-displaceable.

Thomas Kragh (Uppsala)

April 26, 14:15

Room: Ångström 64119

Title: Waldhausen's K-theory of spaces and exact Lagrangian disc

fillings of the Legendrian unknot.

Abstract: I will discus some basic notions around Waldhausen's

definition of algebraic K-theory of spaces. I will then relate this to

generating families quadratic at infinity for any Lagrangian in R^2n

equal to the standard at infinity (previous talks was around the

existence of such generating families - for this talk I will simply

assume these). I will also discus an easier "Morse-type" proof of an

important result by Bökstedt, which relates to the Lagrangian Gauss map

of such Lagrangians. In fact, I will sketch a proof that the Gauss map

relative to infinity is in fact trivial - i.e that any disc filling of

the Legendrian unknot has trivial Gauss map.

April 5, 14:15

Room: Ångström 64119

Title: The simplest open Gromov-Witten invariant is the count of holomorphic Maslov index 2 disks with boundary on a smooth Lagrangian submanifold. I will explain a "refined way" to count such disks on an immersed Lagrangian, focussing on dimension 4. I will talk about the surrounding context of local mirror symmetry, and as a different application, I will exhibit Lagrangian Whitney spheres in CP2 which are Hamiltonian non-displaceable from the complex line. This is joint work in progress with G. Dimitroglou Rizell and T. Ekholm.

March 29, 14:15

Room: Ångström 64119

Title: Symmetries in monotone Lagrangian Floer theory.

Abstract: Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I'll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

February 22, 14:15

Room: Ångström 64119

Title: Spaces of Generating Families and the Hatcher-Waldhausen map.

Abstract: In a previous talk I outlined how to construct a generating

family quadratic at infinity for any Lagrangian L in R2n, which is

standard at infinity (equals Rn). In this talk I will talk about the

consequences of having such a generating family, and about related

classifying spaces. In particular I will show that the Langrangian Gauss

map to U/O is homotopy trivial. This means that in all dimensions not

equal 3 mod 4 the immersion class (relative infinity) is trivial (this

is only news in dimension 4k+1).

February 15, **15:15**

Room: Ångström 64119

Title: Log Geometric Techniques in Mirror Symmetry

Abstract: We will first discuss an algebraic geometric approach to the Fukaya category in symplectic geometry in terms of punctured log Gromov-Witten theory. For this our main object of study is a degeneration of elliptic curves, namely the Tate curve. This is the easiest non-trivial example of a toric degeneration in the Gross-Siebert program concerning mirror symmetry. We will also discuss more general toric degenerations as well as the topology of their real loci. For this we will look at Kato-Nakayama spaces associated to log schemes. This is mostly joint work with Bernd Siebert, some parts based on discussions with Mohammed Abouzaid.

Tobias Ekholm (Uppsala)

February 8, 14:15

Room: Ångström 64119

Title: Legendrian surgery and partially wrapped Floer cohomology II.

January 11, 14:15

Room: Ångström 64119

Title: Monotone Lagrangians in cotangent bundles of spheres

Abstract: We show that there is a 1-parameter family of monotone Lagrangians in cotangent bundles of spheres with the following property: every (orientable spin) closed monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the Lagrangians in the family. The proof involves studying a version of the wrapped Fukaya category that includes monotone Lagrangians. This is joint work with Mohammed Abouzaid.

## 2016

**Tobias Ekholm**(Uppsala)

December 7, 14:15

Room: Ångström 64119

Title: Legendrian surgery and partially wrapped Floer cohomology

**Bruno Martelli**(Università di Pisa)

November 30, 14:15

Room: Ångström 64119

Title: Hyperbolic cone-manifolds in dimension four

Abstract: A hyperbolic (or flat, spherical) cone-manifold is a kind of constant curvature manifold admitting some types of codimension-two "cone singularities". These objects have been studied a lot in dimension 2 and 3, but not much in higher dimensions. In this seminar we introduce the general theory of cone-manifolds (pioneered by Thurston and McMullen) and describe some new examples in dimension four. These examples are interesting because the structure can be deformed in an appropriate way, displaying some new flexibility phenomena in higher-dimensional hyperbolic geometry.

**Johannes Rau**(Universität Tübingen)

November 23, 14:15

Room: Ångström 64119

Title: Real Hurwitz numbers -- a tropical approach

Abstract: The study of Hurwitz numbers, despite its long history, has been completely remodeled in the last twenty years with the discovery of deep connections for example to Gromov-Witten theory and matrix integrals, originating in string theory. While classical Hurwitz numbers count certain holomorphic maps, sometimes it is natural to look at the "real" version of the problem (counting holomorphic maps compatible with a given real structure). I will present a tropical approach (i.e. based on pair-of-pants decomposition) to calculate such real Hurwitz numbers. Another recent development is the construction of a signed invariant count of real maps by Itenberg and Zvonkine (in the spirit of Welschinger invariants). If time permits, I will discuss possible relations with the tropical approach.

Lionel Lang (Uppsala)

November 16, 14:15

Room: Ångström 64119

Title: The vanishing cycles of curves in toric surfaces (joint work with Rémi Crétois)

Abstract: Take a generic curve C in a linear system L on a toric surface X. What are the simple closed curves in C that can be contracted along a degeneration to a nodal curve? This question can be rephrased in term of the image of the monodromy map given by the complement of the discriminant D ⊂ L into the mapping class group of C. Compared with degeneration in M̅g (g>1) where any cycles can be contracted, there are known obstructions to contract cycles in L, namely roots of the relative canonical bundle and hyperelliptic involution. We will review how we can detect them and show that there is no other obstructions. Indeed, we will show that the monodromy is surjective on MCG(C) when no such obstruction appears. If time permits, we will also discuss the image of the monodromy in some obstructed cases.

There will be two main ingredients along the proof. First we will use explicit degenerations on a well studied class of curves: simple Harnack curves. Then we will construct explicit element of the monodromy by applying Mihkalkin's approximation Theorem to well chosen loops in some tropical compactification of the linear system L.

Georgios Dimitroglou Rizell (Uppsala)

November 2, 14:15

Room: Ångström 64119

Title: The classification of Lagrangians nearby the Whitney immersion

Abstract: We classify the Lagrangian tori and spheres in the

four-dimensional vector space that are close (in the appropriate sense)

to Whitney's immersed Lagrangian sphere with a single double point; up

to Hamiltonian isotopy, they are either product tori, Chekanov tori, or

rescalings of the Whitney sphere. While we believe that all Lagrangians

in the vector space can be put in such a position, we also provide

explicit and elementary examples showing that, for tori inside the unit

ball, a significantly larger ball is sometimes needed by the Hamiltonian.

Yankı Lekili (King's College London)

October 26, 14:15

Room: Ångström 64119

Title: Duality between Legendrian and Lagrangian invariants.

Abstract: We will discuss (derived) Koszul duality in the setting of pseudoholomorphic curve invariants

associated to Legendrians and Lagrangians. This is mostly based on a joint work with Ekholm.

Though, I will also allude to some previous results obtained jointly with Etgu.

Evgeny Volkov (Uppsala)

October 12, 14:15

Room: Ångström 64119

Title: On one cyclic A∞ algebra.

Abstract: The de Rham algebra of a manifold is canonically a cyclic A infinity algebra with vanishing higher operations.

We pull this A infinity structure back to the subcomplex of harmonic forms and discuss the question of whether the pull back structure is also cyclic. This is work in progress joint with K. Cieliebak.

Marco Golla (Uppsala)

October 12, 14:15

Room: Ångström 2003 (Note unusual location!)

Title: Signature defects, handles, and ribbon discs

Abstract: The homology groups of a manifold give a lower bound on the number of handles in a handle decomposition (or even on the cells of a CW decomposition). We use Casson-Gordon signatures to improve on this bound for rational homology 4-balls bounding a given rational homology 3-sphere. In turn, this gives information about slice and ribbon discs for knots in the 3-sphere.

This is joint work (in progress) with Paolo Aceto and Ana Lecuona.

Alexander Berglund (Stockholm University)

October 5, 14:15

Room: Ångström 64119

Title: Automorphisms of manifolds and graph homology

Abstract: There is a classical programme for understanding automorphisms of high dimensional smooth manifolds whereby one studies, in turn, the monoid of homotopy automorphisms, the block diffeomorphism group, and finally the diffeomorphism group. The relative homotopy groups in each step are calculated by, respectively, the surgery exact sequence and, in a range, Waldhausen's algebraic K-theory of spaces.

I will talk about the calculation of the stable rational cohomology of the block diffeomorphism group of the g-fold connected sum #^g S^d x S^d relative to a disk (2d>4). Our result is expressed in terms of a certain graph complex, which, quite surprisingly, is related to Kontsevich's graph complex that calculates the cohomology of automorphism groups of free groups. I will also comment on the relation to the results of Galatius and Randal-Williams on the stable cohomology of the diffeomorphism group. This is joint work with Ib Madsen.

Stéphane Guillermou (Université Grenoble Alpes)

September 28, 14:15

Room: Ångström 64119

Title: The three cusps conjecture.

Abstract: Arnol'd's three cusps conjecture is about the fronts of Legendrian

curves in the projectivized cotangent bundle of the 2-sphere. It says

that the front of a generic Hamiltonian deformation of the fiber over a

point has at least three cusps.

We will recall some results of the microlocal theory of sheaves of

Kashiwara and Schapira and see how we can use them to prove the conjecture.

Stefan Behrens (Utrecht University)

September 21, 14:15

Room: Ångström 64119

Title: Surface diagrams and holomorphic quilts.

Abstract: I will report on work in progress with Katrin Wehrheim and Morgan Weiler. The rough idea is as follows. On the one hand, work of Donaldson-Smith and Perutz suggests that suitable counts of "holomorphic multi-sections" of generic maps from 4-manifolds to surfaces should contain information about the Seiberg-Witten invariants. On the other hand, a result of Williams shows the existence of particularly simple maps to the 2-sphere whose structure can be captured by curve configurations in a single fiber, so-called surface diagrams. Combining the two observations, our hope is to eventually find a description of the Seiberg-Witten invariants of 4-manifolds in terms of their surface diagrams.

September 15 (Thursday, at 10:15): Tamas Kalman (Tokyo Institute of Technology)

Title: The Homfly polynomial and Floer homology

Abstract: I will report on a formula that expresses certain extremal coefficients in the Homfly polynomial of an alternating link from the Seifert graph G. This happens in a combinatorially novel way, using the so-called interior polynomial I(G). There is an intermediate step in the computation of I(G) where we consider a particular set of vectors called `hypertrees’. It turns out that hypertrees can be identified with spin-c structures that support a certain sutured Floer homology group. Hence in effect we are computing Homfly coefficients from Floer theory. If time permits, I will speculate on a possible generalization to the non-alternating case.

(I will mention joint results with A. Juhasz, H. Murakami, A. Postnikov, and J. Rasmussen.)

Dmitry Tonkonog (Uppsala)

September 7th, 14:15

Room: Ångström 64119

Title: Laurent phenomenon and symplectic cohomology

Abstract: The function x+y+1/xy stays Laurent under a sequence of certain birational changes of the coordinates x,y called cluster transformations. This phenomenon has been given algebro-geometric meaning through the work of Gross, Hacking and Keel; we will look at it from a mirror-symmetric point of view, through symplectic cohomology.

Paolo Aceto (Rényi Institute)

May 18th, 13:15

Room: Ångström 64119

Title: Knot concordance and homology sphere groups.

Abstract: We study two homomorphisms to the rational homology sphere group. One is the homomorphism from the knot concordance group C defined by taking double branched covers of knots. We show that the kernel of this map is infinitely generated by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls. We also study the inclusion homomorphism from the integral homology sphere group. Using work of Lisca we show that the image of this map intersects trivially with the subgroup generated by lens spaces. As corollaries this gives a new proof that the cokernel of this inclusion map is infinitely generated, and implies that a connected sum of 2-bridge knots is concordant to a knot with determinant 1 if and only if K is smoothly slice.

This is a joint work with Kyle Larson.

Ana Garcia Lecuona (University of Marseille)

May 18th, 14:30

Room: Ångström 64119

Title: Splice links and colored signatures.

Abstract: The splice of two links is an operation defined by Eisenbund and Neumann that generalizes several other operations on links, such as the connected sum, the cabling or the disjoint union. There has been much interest to understand the behavior of different link invariants under the splice operation (genus, fiberability, Conway polynomial, Heegaard Floer homology among others) and the goal of this talk is to present a formula relating the colored signature of the splice of two oriented links to the colored signatures of its two constituent links. As an immediate consequence, we have that the conventional univariate Levine-Tristram signature of a splice depends, in general, on the colored (or multivariate) signatures of the summands. If time permits we will discuss the intricacies of the non generic case.

This is a joint work with Alex Degtyarev and Vincent Florens.

Thomas Kragh

April 27th, 14:05

Room: Ångström 64119

Title: Generating families quadratic at infinity for exact Lagrangians

in R^2n equal to R^n at infinity.

Abstract: In this talk I will describe how Laudenbach and Sikorav

defined a generating family quadratic at infinity for any Lagrangian

Hamiltonian isotopic to the zero section in a cotangent bundle (I will

stay in R^2n for simplicity where the construction was originally due to

Chaperon). I will then use this construction to create a generating

family on contractible fibers with Well-defined fiber-wise Morse theory

for any Lagrangian in R^2n equal to R^n at infinity. I will then explain

how the "with well-defined Morse theory" can be turned into "quadratic

at infinity" in some slightly weaker sense than the strictest sense

imaginable. I will also explain how the strictest sense is different and

how this defines an invariant of the Lagrangian.

April 20th is the day of the Twelth Uppsala Geometry and Physics Seminar.

Johan Björklund

April 13th, 14:15

Room: Ångström 64119

Title: Counting quadrisecants of smooth and real algebraic knots.

Abstract: A quadrisecant of a given knot K is a line intersecting K four times. A generic knot has a finite number of quadrisecants. Pannwitz, Morton and Mond has shown that any nontrivial knot in R^3 possess at least one quadrisecant. This result has been further refined by Kuperberg and Denne.

I will discuss how to use quadrisecants to obtain a smooth isotopy invariant for smooth knots in RP^3 (and show that Pannwitz, Morton and Monds statement does not hold true there) and how to use them to obtain a rigid isotopy invariant (utilizing the complex parts of a real algebraic knot). This is joint work (in progress) with Oleg Viro (Stony Brook).

Anna Sakovich

April 6th, 14:15

Room: Ångström 64119

Title: Mass and center of mass in mathematical general relativity.

Abstract: While the definition of mass and center of mass via the mass density is straightforward in Newton's theory of gravity, the situation in general relativity is more complicated. In this talk we will discuss how to define mass and center of mass of asymptotically Euclidean and asymptotically hyperbolic manifolds, which are important objects in mathematical general relativity arising as hypersurfaces of asymptotically Minkowskian spacetimes. We will focus on geometric aspects of these definitions and discuss some important related results.

Evgeny Volkov

March 9th, 14:15

Room: Ångström 4001

Title: Chain level string topology II

Ralph Morrison (KTH)

March 2nd, 14:15

Room: 64119 Ångström

Title: Bitangents of tropical plane quartic curves

Abstract: A smooth plane quartic curve over an algebraically closed field has 28 bitangent lines. In this talk, I will prove the corresponding result for the tropical world: a tropical smooth plane quartic curve has 7 tropical bitangent lines, up to a natural equivalence. The proof of this result will consider plane tropical curves both as embedded piecewise linear subsets of the Euclidean plane, and as abstract graphs with lengths on the edges. This is joint work with Matt Baker, Yoav Len, Nathan Pflueger, and Qingchun Ren.

Marco Golla

February 17th, 14:15

Room: ITC 6111

Title: Symplectic hats

Abstract: A symplectic cap of a contact 3-manifold Y is a compact 4-manifold with concave boundary -Y; a symplectic hat of a transverse knot K in Y is a compact symplectic surface in a cap, whose boundary is -K.

We study the existence problem for hats and their topology, with a particular emphasis on the case of transverse knots in the standard 3-sphere; we will also discuss applications to fillings of contact 3-manifolds.

This is joint work (in progress) with John Etnyre.

Evgeny Volkov

February 10th, 14:15

Room: Ångström 80101

Title: Chain level string topology I

## 2014

October 8, 13:15

Thomas Kragh

*Exact Lagrangians are simple-homotopy equivalent to the zero-section*

Abstract: In this talk I will describe White-head torsion of finite CW complexes and explain how it is relevant for classification of smooth manifolds. Then I will describe an easy description of the Fukaya-Seidel-Smith spectral sequence and explain how it can be used to prove that any closed exact Lagrangian L in a cotangent bundle is simple homotopy equivalent to the zero section (the last part is joint with M. Abouzaid).

September 24

Kai Cieliebak (Augsburg)

*A remark on the Euler equations of hydrodynamics*

Abstract: The time evolution of an ideal incompressible fluid is described by the Euler equations. In this talk I will discuss a connection between stationary solutions of these equations and symplectic topology, as well as possible applications to questions of hydrodynamic instability.

September 17

Tobias Ekholm

*Generalizations of knot contact homology and colored HOMFLY *

June 25

Emilia Lundberg

*A bar complex in Morse theory and what is it isomorphic to?*

Abstract: I will define a Morse bar complex (or an A-infinity-structure on the Morse complex) originating from a Morse function on a closed Riemannian manifold M by perturbating vector fields. I will also say something about the homology of this complex being isomorphic to the homology of the free loop space of M (work in progress). If time permits I will calculate an example for the n-sphere.

April 2

Cecilia Karlsson

*Orientations of Morse flow trees in Legendrian contact homology*

I will briefly introduce the concept of Morse flow trees in Legendrian contact homology, defined for a Legendrian submanifold $L$ in the 1 jet space of a manifold $M$. Then I will discuss a way of put an orientation on the space of such trees, making it possible to calculate the homology with integer coefficients. This can be done provided $L$ is spin and $M$ is orientable. Using some geometric properties of Morse flow trees, i.e. the stable and unstable manifolds in $M$ corresponding to the gradient flow which define the trees, I will show a way to compute the orientation of rigid Morse flow trees explicitly. This is work in progress.

March 26

Albin Eriksson Östman

*Fully noncommutative Legendrian contact homology with homotopy coefficients, linearizations, 2-copy links, and covers.*

Abstract: We discuss the fully noncommutative Legendrian contact homology with homotopy coefficients of a Legendrian submanifold, L, in a contact manifold, P*\R, where P is an exact symplectic manifold. If the Legendrian contact homology algebra of L can be linearized, then the corresponding linearized complex of the 2-copy link of L contains a subcomplex which can be identified with the Morse complex of the universal cover of L. By reducing the coefficients in the Legendrian contact homology algebra, we get a subcomplex identified with the Morse comlpex of a normal cover of L.

As an example we prove an improved double point estimate for a displaceable exact Lagrangian immersion of the Poincaré homology sphere in an exact symplectic manifold P, provided the Legendrian contact homology algebra of L is linearizable.

February 19

Till Brönnle

*Extremal Kähler metrics on projectivized vector bundles*

Abstract*: *This talk shall be a sequel to my talk last year on the subject of extremal Kähler metrics on projectivised vector bundles. However, we shall focus more on the analytic aspects of the problem and will discuss how to deal with the fully non-linear fourth order PDE which has to be solved in order to construct such an extremal Kähler metric. The major technical difficulty is to control the adiabatic parameter involved in the construction and we shall explain how to deal with this.

## 2013

March 6

Tobias Ekholm

*Constructing Lagrangian immersions with few double points*

Abstract: We describe how Murphy's theory of loose Legendrian submanifolds leads to Lagrangian immersions with surprisingly few double points.

The talk reports on joint work with Elliashberg, Murphy, and Smith.

## 2012

October 31

Tobias Ekholm

*Large N-dualities and knot contact homology III *

Abstract: This is the third talk in a series where we intend to discuss recent observations relating Chern-Simons theory and topological strings to augmentation varieties arising in knot contact homology.

October 17

Tobias Ekholm

*Large N-dualities and knot contact homology II *

Abstract: This is the second talk in a series where we intend to discuss

recent observations relating Chern-Simons theory and topological strings

to augmentation varieties arising in knot contact homology.

September 26

Tobias Ekholm

*Large N-dualities and knot contact homology I *

Abstract: This is the first talk in a series where we intend to discuss recent observations relating Chern-Simons theory and topological strings to augmentation varieties arising in knot contact homology. During the first talk we discuss background material on both the physics a mathematics sides.

September 19

Thomas Kragh (Uppsala)

*Stable homotopy types in Floer theory*

Abstract: I will start by giving a crash course in Morse theory and describe how Morse homology is used to define invariants in symplectic topology. Then I will explain why spaces constructed from Morse theory are stronger invariants than Morse homology, but also why we can never hope to refine Floer homology to a space valued invariant. This naturally leads us to the definition of (pre-)CW-spectra, and I will explain how these are weaker invariants than spaces yet stronger than homology. Finally, I will discuss some cases where one can refine Floer homology into spectrum valued invariants and some results coming from this.

September 5

Reza Rezazadegan (Uppsala)

*On a spectral sequence for Lagrangian Floer holomogy*

Abstract: Fibered Dehn twists are certain symplectomorphisms associated to Lagrangian spheres and more generally spherically fibered coisotropic submanifolds of a symplectic manifold. They generalize the familiar two dimensional Dehn twists. In this talk I outline how the effect of a composition of such Dehn twists on Floer homology can be given as a hypercube of resolutions. An important tool here is quilted Floer homology. Time permitting, I outline how the "spectral sequence of branched double covers" is a special case of the one mentioned above. This spectral sequence implies that by adding extra terms to the Khovanov hypercube of a link L, coming from counting holomorphic polygons in symmetric products of surfaces, one obtains the Heegaard-Floer homology of the branched double cover of L.

May 30

Johan Björklund

*Flexible isotopy classification of flexible knots*

In this talk we will define flexible knots, objects meant to capture the topological properties of real algebraic knots, and then use them to introduce flexible isotopy, that is, an isotopy which is at all times a flexible knot. We will also briefly present Viros encomplexed writhe using Ekholms interpretation in terms of the shade number. It will be shown that two genus 0 flexible knots of degree d are flexibly isotopic if, and only if, their real parts are smoothly isotopic and their encomplexed writhes coincide. If time allows we will also see that there are comparatively "many" flexible knots compared to real algebraic knots of a given degree (considered up to flexible and rigid isotopy respectively).

March 7

Johan Källén (Dept. of Physics and Astronomy)

*Topological field theories and contact structures*

Abstract: I will describe an interesting interplay between contact

geometry and constructions of topological field theories. The focus is

on a recently constructed five dimensional theory, and our formalism

suggest a generalization of the instanton equations to five dimensional

contact manifolds. For the special case of the five manifold being a

circle bundle over a four dimensional symplectic manifold with

integral symplectic form, I will describe how to calculate explicit

expressions for the partition function of the theory using the

technique of path integral localization, which is an

infinite-dimensional generalization of the Atiyah-Bott-Berline-Vergne

localization theorem for finite dimensional integrals. These types of

calculations have received a lot of interest in the case of three

dimensional topological field theories, since it gives a new

perspective on invariants arising from Chern-Simons theory on Seifert

manifolds.

February 22

Tobias Ekholm

*Legendrian knots and exact Lagrangian cobordisms*

Abstract: An exact Lagrangian cobordism between Legendrian links induce a DGA-morphism from the Legendrian algebra of the link at the positive end to that of the link at the negative end. We give a flow tree description of such maps for cobordisms that are Lagrangian submanifolds in a class of symplectic manifolds with ends that are half-symplectization of standard contact 3-space. In particular, for elementary cobordisms, i.e. cobordisms that corresponds to certain modifications of Legendrian knots, this description leads to a purely combinatorial computation of the cobordism map. As an application we construct non-deformation equivalent exact Lagrangian surfaces that fill a fixed Legendrian link. Furthermore, we observe that an exact filling of a Legendrian knot that is composed of certain elementary cobordisms define an element in the Khovanov homology of the knot.

January 26

Emilia Lundberg

*A bar construction in Morse-Witten homology (presentation of master thesis) *

Abstract: We introduce an $A_\infty$-structure on the Morse-Witten complex of a smooth closed manifold $M$, where the operations are defined by counting perturbed Morse flow trees. Conjecturally, the corresponding Hochschild homology is closely related to the singular homology of the free loop space of $M$. We show, by direct calculation, that the two are isomorphic for products of spheres.

## 2011

December 14

Vladimir Chernov (Dartmouth)

*Relations between contact and Lorentz geometry *

Abstract: We show that for many spacetimes causal relation between two points is equivalent to the Legendrian linking of spheres of lights rays through these points in the contact manifold of all light rays. This gives solution to the Low Conjecture and the Legendrian Low conjecture formulated by Natario and Tod. We also discuss causal structure on the space of Legendrian submanifolds in a contact manifold. Finally we explain how globally hypebolicity of the Lorentz metric determines the smooth structure on the spacetime. This leads to the question whether the contact manifold of all light rays determines the spacetime and its Cauchy surface.

November 23

Tobias Ekholm

*Exact Lagrangian immersions with one double point II *

Abstract: We show the following: if $K$ is a $2k$-manifold, $k>2$, with Euler characteristic different from $-2$ that admits an exact Lagrangian immersion into ${\mathbb C}^{2k}$ with one transverse double point and no other self interscetions then $K$ is diffeomorphic to the standard $2k$-sphere. In particular, the result rules out Lagrangian immersions of exotic spheres with only one double point, even though such spheres admit Morse functions with only two critical points. The result discussed is joint work with Ivan Smith.

November 16

Baptiste Chantraine (Université Libre de Bruxelles)

*Bilinearized Legendrian contact homology*

Abstract: Linearisation of Legendrian contact homology is a tool to extract finite dimensional invariants out of the Chekanov algebra of a Legendrian submanifold. A drawback of the construction is that on the first order the theory becomes commutative. In this talk, we will introduce a generalisation of this tool called bilinearisation which keeps track of the non commutativity of the Chekanov algebra even at the first order. This construction also appears to be an effective tool to distinguish augmentations of a Legendrian submanifold. We will provide examples and geometrical interpretations of both of those aspects. This is a joint work with Frédéric Bourgeois.

November 9

Tobias Ekholm

*Exact Lagrangian immersions with one double point *

Abstract: We show the following: if $K$ is a $2k$-manifold, $k>2$, with Euler characteristic

different from $-2$ that admits an exact Lagrangian immersion into ${\mathbb C}^{2k}$ with

one transverse double point and no other self interscetions

then $K$ is diffeomorphic to the standard $2k$-sphere. In particular, the result rules out

Lagrangian immersions of exotic spheres with only one double point, even though such spheres

admit Morse functions with only two critical points. The result discussed is joint work with Ivan Smith.

November 2

Andreas Juhl

*Einstein metrics of negative curvature and conformally invariant differential operators *

Abstract: I will first discuss the notion of Poincare-Einstein metrics in the sense

of Fefferman and Graham. These are asymptotically hyperbolic metrics

which can be considered as real analogs of the Kaehler-Einstein metrics

of Cheng and Yau. In recent years, their study has been much stimulated

by its interaction with the AdS/CFT-duality in physics.

Poincare-Einstein metrics correspond to conformal classes on its

boundary at infinity. This bulk-space/boundary correspondence is the

basis of their usage in conformal differential geometry. Then I will

describe how the correspondence can be used to understand the structure

of conformally invariant powers of the Laplacian (which generalize the

Yamabe operator). Among other things, these operators give rise to

interesting action functionals, lead to the notion of Branson's

Q-curvature and are related to functional determinants. In a sense, the

results can be regarded as incarnations of the duality suggested by

physics.

October 26

Douglas LaFountain (Århus)

*The space of filtered screens and holomorphic curves *

Abstract: For a genus g surface with s> 0 punctures and 2g+s> 2,

decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over

the usual Teichmuller space, where the fiber corresponds to families

of horocycles peripheral to each puncture. As proved by R. Penner,

DTeich admits a mapping class group-invariant cell decomposition,

which then descends to a cell decomposition of Riemann's moduli space.

In this talk we introduce a new cellular bordification of DTeich

which is also MCG-invariant, namely the space of filtered screens.

After an appropriate quotient, we thus obtain a cell decomposition for

a new compactification of moduli space; if time permits we indicate

possible ways this technology may be used to study moduli spaces of

holomorphic curves. This work is joint with R. Penner.

September 14

Clement Hyvrier

*Hamiltonian fibrations and the Weinstein conjecture*

Abstract: Using the relation between the non-vanishing of some Gromov-Witten invariants and the existence of a closed characteristics for any closed separating contact hypersurface due to Hofer and Viterbo, we will in show that the Weinstein conjecture holds for Hamiltonian fibrations over symplectically uniruled spaces, i.e. symplectic spaces for which there is a non vanishing genus zero Gromov-Witten invariant with one point constraint (or, roughly speaking, for which there is a rational curve through any point).

April 20

Justin Pati (Uppsala)

*Convergence to Reeb Chords of Pseudoholomorphic Strips in Symplectizations*

Abstract: We will discuss the assumptions and proofs of several results pertaining to asymptotic behavior of pseudoholomorphic strips.

These results are crucial for the analytic definition of the various versions of the differential in contact homology and Legendrian contact homology. The talk will be reasonably elementary and will hopefully bring out the contact geometric aspects of the pseudoholomorphic curve theory. We will begin with some rather general theory valid on the

symplectization of any compact contact manifold. The main ingredients here are area and energy along with Hofer's bubbling lemma. Convergence to Reeb chords in the chord generic case will follow directly. Next we will try to understand the Morse-Bott case. This will involve a nice choice of coordinates which in fact cover the whole asymptotic story of a strip with a single coordinate chart.

January 26, Licentiate seminar, in Häggsalen (Note the change of room)

Georgios Dimitroglou Rizell

*Knotted Legendrian surfaces with few Reeb chords*

Abstract: For g > 0, we construct g + 1 Legendrian embeddings of a surface of genus g into J1(R2) = R5 which lie in pairwise distinct Legendrian isotopy classes and which all have g +1 transverse Reeb chords (g +1 is the conjecturally minimal number of chords). Furthermore, for g of the g + 1 embeddings the Legendrian contact homology DGA does

not admit any augmentation over Z2, and hence cannot be linearized.

We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in J1(S2) from a similar perspective.

January 19, Thesis presentation

Cecilia Karlsson

*Area preserving isotopies of self transverse immersions of S1 in R2*

## 2010

Wednesday, December 15

Ezra Getzler (Northwestern University)

*A Filtration of Open/Closed Topological Field Theory*

Wednesday, November 24

Ramon Horvath (Uppsala)

*On A∞ algebras II*

Wednesday, November 17

Ramon Horvath (Uppsala)

*On A∞ algebras*

Wednesday, November 3

Albin Eriksson Östman (Uppsala)

*A smooth isotopy invariant defined through relative contact homology*

Wednesday, October 20

Johan Björklund (Uppsala)

*Legendrian contact homology in the product of a punctured Riemann surface with the real line*

Wednesday, October 13

Georgios Dimitroglou Rizell (Uppsala)

*Knotted Legendrian surfaces with few Reeb Chords*

Wednesday, October 6

Tobias Ekholm (Uppsala)

*Legendrian surgery and the symplectic homology product II*

Wednesday, September 29

Tobias Ekholm (Uppsala)

*Legendrian surgery and the symplectic homology product*

Wednesday April 21

Speaker: Yacin Ameur

Title: *The Coulomb plasma*

Wednesday April 14

Speaker: Dorin Cheptea (Romanian Academy)

Title: *The universal finite-type invariant of three-dimensional manifolds and of cobordisms, and applications.*

Wednesday April 7

Speaker: Tobias Ekholm (Uppsala)

Title: *Knot contact homology II*

Wednesday March 31

Speaker: Tobias Ekholm (Uppsala)

Title: *Knot contact homology*

Wednesday March 17

Speaker: Jens Fjelstad (Karlstad)

Title: *Some properties of quantum representations of mapping class groups*.

Wednesday March 3

Speaker: Johan Björklund (Uppsala)

Title: *A new invariant for generic real algebraic surfaces in RP³*

Wednesday February 17

Speaker: Tobias Ekholm (Uppsala)

Title: *Legendrian homology – S³ vs R³*

Wednesday February 3

Speaker: Daniel Mathews

Title: *Chord diagrams, contact-topological quantum field theory, and contact categories*